For requesting, clarifying, and comparing definitions of mathematical terms.

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1answer
16 views

Definition of $\{t\in T: X(t) = u\}$, where $\{X(t): t\in T\}$ is a stochastic process

Maybe this is a silly question, but it's important that I understand this. I just started to read the book Level Sets and Extrema of Random Processes and Field, from Jean-Marc Azais and Mario ...
0
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1answer
42 views

difference between a string and alphabet symbols

What's the difference between a string and a symbol in the alphabet for an automata? For example if you have an alphabet $\Sigma={0,1}$ Is a string a particular combination (set?) of alphabet ...
0
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0answers
26 views

Where does (remainder)/$(\Delta t)^2$ come from?

Can someone please explain what is going on here? Where does (remainder)/$(\Delta t)^2$ come from? How did we go from $(1/2)\|\boldsymbol{\ddot\gamma(t)}\|(\Delta t)^2$ to just ...
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5answers
54 views

Definition for supremum

Is the below definition for supremum correct? If no, then how to define it in similar way? $$\sup E = s \Longleftrightarrow(\forall t, \; \forall x, \; x \le t \Longrightarrow x \le s \le t) $$ We ...
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1answer
23 views

Should you drop the inner absolute value sign for $L2$ norm?

Lp norm is defined as: $ \left\| \mathbf{x} \right\| _p := \bigg( \sum_{i=1}^n \left| x_i \right| ^p \bigg) ^{1/p}$ But often time I see people writing: $\left\| \mathbf{x} \right\| _2 := \bigg( ...
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7answers
832 views

Question about the derivative definition

The derivative at a point $x$ is defined as: $\lim\limits_{h\to0} \frac{f(x+h) - f(x)}h$ But if $h\to0$, wouldn't that mean: $\frac{f(x+0) - f(x)}0 = \frac0{0}$ which is undefined?
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3answers
205 views

Vacuous Truth and Universal Conditional Statements

Sometime after I began studying conditional statements, I started having difficulty understanding vacuous truth. For instance, the fact that for any set $A$ we have $\emptyset\subset A$ is commonly ...
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1answer
38 views

Skeptical about (my understanding of) wikipedia's definition of a reflective subcategory.

I am self-learning category theory (though, at this point, I no longer remember what got me started), and I have encountered a troubling definition on wikipedia. The formal definitions of (full) ...
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1answer
26 views

Definition of a Hilbert basis

Given a Hilbert space $\cal H$, what criterion describes the property "$\cal B$ is a Hilbert basis for $\cal H$"? It would be even better if the definition can be stated in a way that mimics some ...
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0answers
23 views

Define set of all twin primes without mention of prime number?

Do we have a definition of a set that is equivalent to the set of all twin primes such that the definition does not make mention of a prime number? If not, would the discovery of such a definition be ...
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1answer
40 views

Question About Definition of Lyapunov Exponents

I just have a quick question about the definition of Lyapunov exponents. My textbook defines them for a smooth map $f:M\to M$, where $M$ is a smooth manifold. For $x\in M$ and $v\in T_xM$: ...
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1answer
72 views

Is $\sqrt{90x^2}$ fully simplified?

Is the following fully simplified? $$\sqrt{90x^2}$$ My math teacher gave me two points less because of this. The right solution is $\sqrt{90}|x|$ he thinks.
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0answers
25 views

How is a characteristic subgroup verified?

I know the definition of a characteristic subgroup: $\sigma (H)=H$ for all $\sigma \in \text{aut}\, G$ where $H \leq G$. But, I do not understand how $\sigma$ is defined. Surely we can map $H$ to $H' ...
0
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0answers
14 views

Is there a relation between Entropy number and Minkowski–Bouligand dimension

I was trying to have a basic understand of entropy number; but when I tried with Wikipedia; it got directed to Minkowski–Bouligand dimension, as shown in the link. Is there a relation between them? ...
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8answers
99 views

How is $i$ defined, really? And why is $i \neq -i$?

$i$ is generally defined as $\sqrt{-1}$ which is ambiguous because $\sqrt{x}$ is defined as the positive number whose square is $x$; however $i$ can't be positive since it isn't real. Ok, so what if ...
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1answer
41 views

definition of the notion of a subcategory of an $\infty$-category

Is the following formulation of the notion of a subcategory of an $\infty$-category ( - in the sense of section 1.2.11 of Lurie's Higher Topos Theory - ) correct?: Remark. Let $\mathcal{C}$ be an ...
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1answer
25 views

Clearing the concept of definition

What is the definition in general? This question has arised mainly due to the fact that the word "sphere" is used for 3-dimensional space. What if I want to define for any space? For a real line or ...
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2answers
87 views

What do mathematicians mean by if two vector spaces are isomorphic, then “they are the same”

I have been reading some of the answers here that if two vector fields are isomorphic, then they are "essentially the same". For example, an answer here:What does "isomorphic" mean in ...
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0answers
10 views

What does “smoothly dependent” mean here?

Let $T\colon Y\times M\to Y$ be an operator, where $M$ is the parameter space and $Y$ is a normed space. $T$ is called smooth on $Y$ if both map $T$ and its derivative $T'(y)$ are ...
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0answers
62 views

What's the intuition behind the direct integral of a family of Hilbert spaces?

In order to understand better the mathematically rigorous version of Dirac's formalism in Quantum Mechanics I've been reading about direct integrals of Hilbert spaces, projector-valued measures and so ...
0
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1answer
22 views

What's the usual convention for directional derivatives in the $0$ direction?

Depending on the source, the definition of the directional derivative does not include the restriction that the direction vector be of unit length. In this case, it seems to me that we can then in ...
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1answer
39 views

Compatibility of group structure and topological structure for topological groups

I am fairly new to the concept of topological groups, and would like to understand the underlying idea. My question is about the compatibility between the two structures. The definition of a ...
2
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0answers
21 views

Definition of Saturating set

Notations:$ \bar{G}(A)$ denotes the subgraph induced by $A$ in the complement of $G$. Definition see definition 2.15: Given a set $s$ of vertices, an edge $e$ of $G(V − s)$ is said to be ...
4
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1answer
41 views

Measures which are not real-valued

I'm currently starting to study measure theory and the definitions I've seem up to now are: A measurable space is a pair $(X,M)$ being $X$ a non empty set and $M$ a $\sigma$-algebra of subsets of ...
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0answers
38 views

What does maximal order mean in a group?

As written in Abstract Algebra by T. W. Judson: Lemma 13.4 : Let $G$ be a finite abelian $p-$group and suppose that $g ∈ G$ has maximal order. Then $G$ is isomorphic to $g × H$ for some subgroup ...
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1answer
26 views

Is a non-principal ultrafilter the same thing as a free ultrafilter?

Can someone please confirm if a non-principal ultrafilter is the same thing as a free ultrafilter. I keep finding conflicting definitions so am not sure.
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1answer
50 views

The domain of the product of two functions [duplicate]

It is well known that the function $f(x)=1/x$ is not defined for $x=0$. However, simply multiplying $f$ by the function $g(x)=x$ gives a constant, very well defined, function, even at $x=0$. How can ...
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2answers
65 views

The equivalence of open-set definition of continuous mappings to the $\epsilon-\delta$ definition

A function $f: M\rightarrow N$ is defined to be continuous iff $\forall$ open set $U\subseteq N$, the preimage of $U$ is also open. I was trying to prove that such a definition would be equivalent to ...
1
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1answer
75 views

What is randomness?

It is very to give problems of the sort: I have a set $S$ and all elements of $S$ is a certain value assigned. For example it is very easy to ask, how likely is it, that a thrown common, fair dice ...
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1answer
31 views

What is the definition of sequential precompactness?

I know that topological space $X$ is called precompact if any sequence in $X$ has a subsequence convergent in X. In my book of calculus of variation I have encountered the word sequential ...
2
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0answers
25 views

Clarify the definition of a transition matrix

I am reading the book Matrix Variate Distribution by A. K. Gupta and D. K. Nagar. In the first chapter (Definition 1.2.8), they define a matrix $B_{p}$ ($p \in \mathbb{N}^{\ast}$) as follows : ...
2
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2answers
54 views

Basis-Independent Definition of Gradient of Multivector Fields

The definition of the gradient operator on multivector fields in geometric calculus seems to be $$\nabla = \sum_i e_i\partial_i$$ where $\{e_i\}$ is an orthonormal basis. That's useful, but it's ...
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8answers
397 views

$\epsilon - \delta$ definition of a limit?

The definition in my book is as follows: Let $f$ be a function defined on an open interval containing $c$ (except possibly at $c$) and let $L$ be a real number. The statement $$\lim_{x \to c} f(x) ...
1
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1answer
43 views

Why is the derivative used to represent marginal cost instead of the difference?

Marginal cost is informally defined as "the change in the total cost that arises when the quantity produced is incremented by one unit." And given a total cost function $C(q)$ that's differentiable, ...
0
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1answer
25 views

Proving definition of directional derivative $\lim_{s \to 0} \frac{f(a + sx) - f(a)}{s} = f'(a)x$

Inspired by: Directional derivative question I had long sought for a proof $$\lim_{s \to 0} \frac{f(a + sx) - f(a)}{s} = f'(a)x$$ But whenever I look it up, I always get some multivariable calculus ...
5
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2answers
90 views

What does it mean to 'preserve the first fundamental form'?

I'm a bit confused about the phrase 'preserving the first fundamental form', or 'The Gaussian curvature is determined by the first fundamental form'. For example, let's say I have two surfaces $M$ ...
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0answers
60 views

Why not generalize the Intermediate Value Theorem

So I know the IVT says that given a continuous function on a closed interval [a,b] then if $f(a) < \gamma <f(b)$ then there is $c\in(a,b)$ such that $f(c) = \gamma$. Is there any reason for not ...
1
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2answers
18 views

Definition for not divergent

What is the definition of real numbers $\{x_n\}$ does not diverge to -$\infty$? Would it be $x_n$ does not go to infinity if and only if there exist $M>0$ for all $N \in \mathbb{N}$ numbers such ...
0
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0answers
39 views

Tricky notation: Need help formulating an expression to define a recursive function involving substitutions

I'm having a difficult time trying to come up with an inductive definition for a function I'm calling $f_i(k)$ in terms of constants $\rho$, $d$, the $1 \times n$ vector $q$, and a $n \times n$ matrix ...
2
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1answer
75 views

On the terms “nowhere dense” and “dense-in-itself”

Are there literal interpretations of the terms "nowhere dense" and "dense-in-itself" from which these terms' definitions follow? If I were to guess what it means for a subset $A$ of a topological ...
1
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1answer
78 views

Can you choose -1 as the multiplicative unit? And what is a positive number?

If one starts with the cyclic group of integers and want to introduce multiplikation the ordinare choice of multiplicative identity is the generator 1. But since 1 and its inverse -1 is sort of the ...
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6answers
1k views

Definition of “simplify”

In mathematics the word "simplify" is used a lot. In a lot of cases it is obvious what actually makes an expression simpler, but not always. Is there a measurable definition of simplicity or is it ...
1
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1answer
100 views

Why a linearly independent set of vectors must not contain the zero vector?

Why is it necessary for a linearly independent set of vectors to not contain the zero vector? I am looking with the definition perspective i.e. why do we define linear independence in this way?
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0answers
37 views

Euclidean Neighbourhoods Retracts and Deformation Retracts

Aim of this question is to clarify the differences between the two concepts written in the title, because it's unclear to me wether one condition implies the other, under which circumstances they are ...
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0answers
65 views

Cluster Points of $S_{\Omega}$

Let $T: X \to X$ a continuous map on a compact set $X\subset \mathbb{R}$. A point $x \in X$ is non-wandering if for any open set $U \ni x$ there exists $n>0$ such that $T^{n}(U)\cap U \ne ...
0
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1answer
46 views

What is a smooth function?

According to Wikipedia, a smooth function is a function that has derivatives of all orders. I don't understand what this means if the case was for example the function $$f(x) = 1+2x$$ This can be ...
1
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1answer
26 views

Relation between two different definitions of quadratic convergence

I'm trying to prove quadratic convergence of the multivariate Newton-Raphson method, and I've proved that for some constant $C>0$, we have $\lVert x_{k+1}-a \rVert \leq C \cdot \lVert x_k - a ...
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2answers
112 views

Demystifying definition of the Random Variable

The intro on Random Variables says that it is a variable (in bold), whose value depends on a chance. IMO, it sounds like a random value generator, whose value depends on a chance, just as random ...
4
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1answer
63 views

entire functions and multi-valued functions, an easy to understand explanation?

From wikipedia: The Bessel function of the first kind is an entire function if α is an integer, otherwise it is a multivalued function with singularity at zero. I have plotted the ...
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0answers
9 views

Explanation of defintion of packing number

$X$ is a ground set, and $\mathcal F$ is a system of sets on $X$. Packing/matching number of $\mathcal F$ is defined as: $\nu(\mathcal F) = \sup\{|\mathcal M|: \mathcal M \subseteq \mathcal F, M_1 ...